Remark 4 Note here that for the case of identical diffusion coefficients, i.e., \(d_1=\ldots =d_m\) , we might solve the problem by static feedback instead of the dynamic law of Theorem REF . In that case, system is stabilizable in accordance with boundary controllability studies (see [1]}) as a consequence of the identical diffusion coefficients even if we omit Assumption REF . Indeed, by performing modal decomposition \(z(t,x)=\sum _{n=1}^{+\infty }z_n(t)\varphi _n(x)\) , we would obtain the following ODE system for the first \(N\)  modes: \( \dot{Z}(t)=A Z(t)+\tilde{B} u(t)\)  with \(Z:=\left(z_1,\ldots , z_N\right)^\top \) , \(A\)  given by (REF ) and \(\tilde{B}= \begin{pmatrix} \tilde{B}_1 \\ \vdots \\ \tilde{B}_N \end{pmatrix} ;\tilde{B}_n=d_m\left( (1-\gamma _2)\varphi _n(L)-\gamma _2\varphi _n^\prime (L)\right)B \begin{pmatrix}1&\cdots &1\end{pmatrix}.\)  Then, since \(D=d_mI_m\) , the eigenvalues of \(A\)  are distinct and by the Hautus lemma, system \(Z\)  is stabilizable, whereas for distinct diffusion coefficients, Hautus test would fail. We can, hence, choose \(u_j(t)=K_jz^N(t), j=1\ldots ,N\) , where \(K_j\)  are retrieved by an inequality of the form (REF ) in Section . Then, we can follow similar procedure as in the proof of Theorem REF  to show stability of the closed-loop system. However, contrary to the method of Theorem REF  on internal stabilization, we would here need to solve an LMI of dimension \(mN\times mN\) .

